JEE Main · 2023 · 11 Apr 2023 · Shift 1 · Hyperbola

For $H_n$, $a^2=1+n$ and $b^2=3+n$, so $e^2=1+\frac{b^2}{a^2}=\frac{2n+4}{n+1}=\frac{2(n+2)}{n+1}$. For even $n$ the numerator $2(n+2)$ and denominator $n+1$ are coprime, so $e$ is rational only when both are perfect squares. For even $n$, $n+1$ is an odd perfect square at $n=8,24,48,\dots$; the first of these for which $2(n+2)$ is also a perfect square is $n=48$, where $2(n+2)=100$. Hence $k=48$, giving $a^2=49$, $b^2=51$ and $e=\frac{10}{7}$. The length of the latus rectum is $l=\frac{2b^2}{a}=\frac{102}{7}$, so $21l=306$.